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We investigate decoupling, one of the most important primitives in quantumShannon theory, by replacing the uniformly distributed random unitariescommonly used to achieve the protocol, with repeated applications of randomunitaries diagonal in the Pauli-$Z$ and -$X$ bases. This strategy was recentlyshown to achieve an approximate unitary $2$-design after a number ofrepetitions of the process, which implies that the strategy gradually achievesdecoupling. Here, we prove that even fewer repetitions of the process achievedecoupling at the same rate as that with the uniform ones, showing that ratherimprecise approximations of unitary $2$-designs are sufficient for decoupling.We also briefly discuss efficient implementations of them and implications ofour decoupling theorem to coherent state merging and relative thermalisation.

Decoupling with random diagonal unitaries